CN111475904A - Method for calculating alternating current loss of low-temperature superconducting magnet - Google Patents
Method for calculating alternating current loss of low-temperature superconducting magnet Download PDFInfo
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Abstract
The invention provides a method for calculating AC loss of a low-temperature superconducting magnet, which comprises the following steps: establishing a model of a low-temperature superconducting coil to be detected; setting constraint conditions of the model; solving the model based on the constraint includes: acquiring the magnetic field amplitude and the magnetic field change rate of the model at any moment based on an H equation, and acquiring the maximum value of the magnetic field amplitude of each turn of superconducting strand of the model in one period and the function of the magnetic field change rate and time based on the magnetic field amplitude and the magnetic field change rate; obtaining the critical current and the critical current density of each turn of the superconducting strand according to the maximum value of the magnetic field of each turn of the superconducting strand; acquiring total coupling loss according to the function of the magnetic field change rate and time of each turn of superconducting strand and the coupling time constant; and acquiring the total hysteresis loss according to the function of the magnetic field change rate and the time of each turn of the superconducting strand of the model in one period, the critical current and the critical current density. The method can solve the technical problems of high cost, high difficulty, complex test and the like of the conventional low-temperature superconducting magnet alternating current loss test method.
Description
Technical Field
The invention relates to the technical field of superconducting power, in particular to a method for calculating alternating current loss of a low-temperature superconducting magnet.
Background
With the development of society and the progress of science and technology, people have higher and higher requirements on the efficiency and energy conservation of electric power equipment. Traditional power equipment is mostly by the copper coil who forms with the copper wire coiling, because the copper line has the resistance for a lot of electric energy transformation heat energy consumption is in the copper coil, thereby makes traditional power equipment's loss great, leads to power equipment's efficiency to reduce. The electric power equipment adopting the superconducting material without resistance and high current-carrying capacity as the coil is considered as a breakthrough of a new technology in the power industry due to the advantages of low loss, large capacity, small size and the like, and has great economic and social benefits.
Superconducting materials are mainly classified into low-temperature superconducting materials and high-temperature superconducting materials. The low-temperature superconducting material is mature at present, has low price, but has high refrigeration cost; high temperature superconducting materials are the hot spot of research in recent years, and have low refrigeration cost and high temperature margin, but the production cost is very expensive due to low yield of long-line production. Therefore, the low-temperature superconducting material has great advantages in practical engineering application and commercialization.
The low-temperature superconducting magnet has zero resistance and no loss in the case of direct current transmission, but when the low-temperature superconducting magnet carries alternating current or is in an alternating electromagnetic field, the superconductor shows a certain electromagnetic energy loss, namely alternating current loss (including hysteresis loss and coupling loss) due to the fact that magnetic flux moves first and is subjected to a flux pinning effect. The alternating current loss is one of important indexes of the low-temperature superconducting power device, is one of important factors influencing the stability of the superconducting power device, and has decisive influence on the refrigeration power and the refrigeration efficiency of the superconducting power device. Therefore, analyzing and quantifying the ac loss of the low-temperature superconducting magnet is a key problem to be solved first in analyzing and designing superconducting power equipment.
At present, the alternating current loss of the low-temperature superconducting magnet is mainly analyzed by adopting a calorimetry method, wherein the calorimetry method is a method for directly measuring the flow of gas volatilized from a low-temperature refrigeration medium (liquid helium) due to the heating of the alternating current loss to measure the alternating current loss, namely, the alternating current loss of the low-temperature superconducting magnet can be obtained according to the gas flow rate and the volatilization amount measured by a gas flowmeter and the latent heat of the low-temperature medium. However, since the refrigeration medium of the low-temperature superconductor is liquid helium, a large amount of liquid helium is needed when the alternating current loss of the low-temperature superconducting magnet is measured, the liquid helium is high in cost, rare resources are available, and the experimental test platform is difficult to build and is not suitable for scheme demonstration and scheme design in the early stage of low-temperature superconducting power equipment. In addition, because of heat leakage of the superconducting magnet system, mechanical loss generated by mechanical vibration after excitation of the superconducting coils and the like, the loss measured by the calorimetry is not only ac loss but also total loss, so that the measurement result is not accurate, and the method cannot provide detailed data support for research on the superconducting ac loss because only the total loss can be provided.
Disclosure of Invention
The invention provides a method for calculating the alternating current loss of a low-temperature superconducting magnet, which can solve the technical problems of high cost, high difficulty, complex test and the like of the conventional method for testing the alternating current loss of the low-temperature superconducting magnet.
The technical solution of the invention is as follows: the method for calculating the alternating current loss of the low-temperature superconducting magnet comprises the following steps:
establishing a model of a low-temperature superconducting coil to be detected;
setting constraints of the model, wherein the constraints comprise: a current function and a magnetic field function;
solving the model based on the constraint condition to obtain the magnetic field amplitude of the model and the function of the magnetic field change rate and time, comprising the following steps:
(1) acquiring the magnetic field amplitude and the magnetic field change rate of the model at any moment based on an H equation;
(2) acquiring the maximum value of the magnetic field amplitude of each turn of superconducting strand of the model in one period and a function of the magnetic field change rate and time on the basis of the magnetic field amplitude and the magnetic field change rate;
obtaining the critical current and the critical current density of each turn of superconducting strand according to the maximum value of the magnetic field of each turn of superconducting strand of the model in one period;
acquiring total coupling loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period and a coupling time constant of the corresponding superconducting strand; acquiring total hysteresis loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period, critical current and critical current density;
and obtaining the model alternating current loss according to the total coupling loss and the total hysteresis loss.
Further, the obtaining of the magnetic field amplitude and the magnetic field change rate of the model at any time based on the H equation includes: acquiring the magnetic field intensity of the model at any moment based on the H equation; obtaining the magnetic field amplitude and the magnetic field change rate at any moment based on the magnetic field intensity at any moment; wherein the H equation is shown as follows:
wherein HxAnd HyThe components in the x direction and the y direction of the magnetic field intensity of the superconducting coil model are respectively, and E and J are respectively the electric field distribution and the current density distribution of the superconducting coil model; mu.s0And murRespectively, vacuum permeability andrelative magnetic permeability; n is the value of n for the superconducting strand.
Further, obtaining the maximum value of the magnetic field amplitude of each turn of the superconducting strand of the model in one period and the function of the magnetic field change rate and time based on the magnetic field amplitude and the magnetic field change rate specifically comprises:
determining an extraction point of any superconducting strand, wherein the extraction point is the closest point on the superconducting strand to the symmetry axis of the model;
and obtaining the maximum value of the magnetic field amplitude at the corresponding extraction point of each turn of the superconducting strand in one period and the function of the magnetic field change rate and time.
Further, obtaining the critical current and the critical current density of each turn of the superconducting strand according to the maximum value of the magnetic field of each turn of the superconducting strand of the model in one period, specifically as follows:
for NbTi superconducting strands, determining the relationship between the critical current density of each turn of superconducting strand and the magnetic field and the temperature; or, for Nb3The Sn superconducting strand determines the relation between the critical current density of each turn of superconducting strand and the magnetic field, the temperature and the stress;
and calculating the critical current density and the critical current of each turn of the superconducting strand based on the determined relation.
Further, the superconducting strand is an NbTi superconducting strand, and the relationship between the critical current density and the magnetic field and the temperature of each turn of the superconducting strand is shown as the following formula:
wherein, C0，BC20，TC0α, gamma are the first, second, third, fourth, fifth and sixth intrinsic parameters respectively associated with the NbTi superconducting strand structure, T0The temperature at which the low temperature superconducting coil is located; b is the magnetic field of each turn of superconducting strand of the in-period modelA maximum value; j. the design is a squarec(B,T0) Is the critical current density, Jc(B,T0) The ratio of the critical current to the cross-sectional area S of the superconducting strand is defined as the critical current.
Further, obtaining the total coupling loss according to the function of the magnetic field change rate of each turn of the superconducting strand of the model in one period and the time and the coupling time constant of the corresponding superconducting strand, comprising:
obtaining the coupling time constant of each turn of the superconducting strand as shown in the following formula:
obtaining the coupling loss of any superconducting strand in a period based on the coupling time constant of each turn of superconducting strand and the function of the magnetic field change rate and the time of each turn of superconducting strand of the model in the period, as shown in the following formula:
obtaining the total coupling loss of the superconducting coil based on the coupling loss of any superconducting strand as follows:
Qct＝∑Qci×SLi(10)
wherein θ is a coupling time constant; dBiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; mu.s0Is a vacuum magnetic conductivity; p is the torque length of the superconducting strand; rhotIs the effective lateral resistivity of the copper matrix; qciCoupling loss of the ith turn of superconducting strand; qctTotal coupling loss of superconducting coil, cross-sectional area of superconducting strand, LiIs the length of the ith turn of superconducting strand; t is the cycle duration.
Further, obtaining the total hysteresis loss according to the function of the magnetic field change rate of each turn of the superconducting strand of the model in one period and the time, the critical current and the critical current density comprises the following steps:
obtaining the hysteresis loss of any superconducting strand in a period, as shown in the following formula;
obtaining the total hysteresis loss of the superconducting coil based on the hysteresis loss of any superconducting strand in one period as follows:
Qht＝∑Qhi×SLi(14)
wherein Q ishiHysteresis loss, dB, of the ith turn of superconducting strandiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; i isciAnd JciCritical current and critical current density of the ith turn of superconducting strand under corresponding magnetic fields respectively; qhtTotal hysteresis loss of superconducting coil, S cross-sectional area of superconducting strand LiThe length of the ith turn of superconducting strand.
Further, the establishing of the model of the low-temperature superconducting coil to be tested includes:
acquiring electrical and structural parameters of the low-temperature superconducting coil to be detected, wherein the electrical and structural parameters comprise:
determining the type of the superconducting strand used by the low-temperature superconducting coil to be detected, and the self-field critical current and the n value of the superconducting strand; and the structural parameters of the low-temperature superconducting coil to be detected comprise: the number of turns of the coil, the turn-to-turn insulation and the overall dimensions of the coil;
and establishing a model of the low-temperature superconducting coil to be detected in finite element analysis software according to the electrical and structural parameters of the low-temperature superconducting coil to be detected.
Further, setting the constraints of the model comprises:
analyzing the waveforms of the current and the magnetic field faced by the low-temperature superconducting coil to be detected;
and expressing the current and magnetic field waveforms or the waveforms after the current and magnetic field waveforms are simplified by a functional expression and inputting the waveforms into finite element analysis software.
By applying the technical scheme, the method for calculating the alternating current loss of the low-temperature superconducting magnet is provided, and by establishing a model of the low-temperature superconducting coil to be measured and providing constraint conditions of the model, on the basis, the method has the key points that: reasonably extracting and calculating used intermediate parameters through an H equation, namely obtaining the magnetic field amplitude of each turn of folded wire of the low-temperature superconducting coil and a function of the magnetic field change rate and time, and selecting a specific magnetic field amplitude in a period to calculate the critical current and the critical current density; and finally accurately acquiring the alternating current loss based on the alternating current loss. The method provided by the embodiment of the invention can accurately calculate the coupling loss and the hysteresis loss of the low-temperature superconducting magnet under any current and any magnetic field respectively and calculate the total alternating current loss of the low-temperature superconducting magnet only by a designed finite element simulation method, does not need any other external resource, has low cost, is simple and easy to implement, can simultaneously divide the alternating current loss of the low-temperature superconducting magnet under different input conditions by parallel calculation, and has engineering application prospect.
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The accompanying drawings, which are included to provide a further understanding of the embodiments of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention. It is obvious that the drawings in the following description are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort.
Fig. 1 is a flow chart illustrating a method for calculating ac loss of a cryogenic superconducting magnet according to an embodiment of the present invention;
FIG. 2 is a schematic diagram illustrating a magnetic field magnitude and magnetic field change rate extraction point provided in accordance with an embodiment of the present invention;
FIG. 3 illustrates an EAST poloidal field coil (PF) current waveform provided in accordance with an embodiment of the present invention;
fig. 4 is a graph comparing ac loss results of a PF coil of an EAST apparatus provided in accordance with an embodiment of the present invention.
Detailed Description
It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict. The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
The relative arrangement of the components and steps, the numerical expressions and numerical values set forth in these embodiments do not limit the scope of the present invention unless specifically stated otherwise. Meanwhile, it should be understood that the sizes of the respective portions shown in the drawings are not drawn in an actual proportional relationship for the convenience of description. Techniques, methods, and apparatus known to those of ordinary skill in the relevant art may not be discussed in detail but are intended to be part of the specification where appropriate. In all examples shown and discussed herein, any particular value should be construed as merely illustrative, and not limiting. Thus, other examples of the exemplary embodiments may have different values. It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, further discussion thereof is not required in subsequent figures.
As background art relates, there is a disclosure of testing of ac losses in low temperature superconducting magnets (cryosuperconducting coils), primarily calorimetry, which has certain drawbacks, as noted in the background. In addition, the prior art also discloses a large number of methods for measuring the ac loss of the high-temperature superconducting magnet, wherein the ac loss of the high-temperature superconducting magnet is obtained by simulation analysis calculation, for example, the current research on the ac loss of the high-temperature superconducting strip and the coil usually adopts an H equation, etc., because the high-temperature superconducting strip has a simple material structure and belongs to a multi-stack structure, the coupling loss can be ignored, i.e., the H equation can be used for calculation (aiming at obtaining the electric field intensity, etc. and integrating) to obtain the hysteresis loss. However, the method is not suitable for calculating the alternating current loss of the low-temperature superconducting magnet. As is known, a low-temperature superconducting coil is formed by winding a low-temperature superconducting strand, and the low-temperature superconducting strand is formed by twisting hundreds of superconducting filaments in a copper matrix, and the structure of the low-temperature superconducting coil is very complex, so that the alternating current loss of the low-temperature superconducting magnet is only hysteresis loss, and the coupling loss generated by the interaction between the superconducting filaments and the copper matrix is even more important, and therefore, the hysteresis loss of the low-temperature superconducting magnet cannot be calculated and the coupling loss cannot be calculated by only adopting a calculation method of the high-temperature superconducting magnet. Based on the above background, the embodiment of the present invention provides a method for calculating ac loss of a low-temperature superconducting magnet, which includes:
as shown in fig. 1, according to an embodiment of the present invention, there is provided a method for calculating ac loss of a cryogenic superconducting magnet, the method including the following steps:
(1) acquiring the magnetic field amplitude and the magnetic field change rate of the model at any moment based on an H equation;
(2) acquiring the maximum value of the magnetic field amplitude of each turn of superconducting strand of the model in one period and a function of the magnetic field change rate and time on the basis of the magnetic field amplitude and the magnetic field change rate;
step 5, acquiring total coupling loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period and a coupling time constant of the corresponding superconducting strand; acquiring total hysteresis loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period, critical current and critical current density;
and 6, obtaining model alternating current loss according to the total coupling loss and the total hysteresis loss.
In the embodiment of the present invention, in order to obtain a model of the low-temperature superconducting coil to be measured, the following method may be adopted for establishing the model of the low-temperature superconducting coil to be measured:
(1) acquiring electrical and structural parameters of the low-temperature superconducting coil to be detected, wherein the parameters comprise:
determining the type of the superconducting strand used by the low-temperature superconducting coil to be detected, and the self-field critical current and the n value of the superconducting strand; and the structural parameters of the low-temperature superconducting coil to be detected comprise: the number of turns of the coil, the turn-to-turn insulation and the overall dimensions of the coil;
(2) and establishing a model of the low-temperature superconducting coil to be detected in finite element analysis software according to the electrical and structural parameters of the low-temperature superconducting coil to be detected.
The above-mentioned electric and structural parameters of the low-temperature superconducting coil to be measured can be obtained by adopting the conventional technical means in the field.
It should be understood by those skilled in the art that, based on the known electrical and structural parameters of the low-temperature superconducting coil to be tested, modeling the low-temperature superconducting coil to be tested in finite element analysis software is a technique known in the art and will not be described in detail herein.
Wherein it can be modeled in the finite element analysis software COMSO L.
In this embodiment of the present invention, in order to provide a constraint condition for the model, setting the constraint condition of the model may include:
(1) analyzing the waveforms of the current and the magnetic field faced by the low-temperature superconducting coil to be detected;
(2) and expressing the current and magnetic field waveforms or the waveforms after the current and magnetic field waveforms are simplified by a function expression and inputting the waveforms into finite element analysis software.
In order to obtain the ac loss of the low-temperature superconducting coil to be measured, the electromagnetic environment of the superconducting coil needs to be analyzed, such as the current waveform passed through the superconducting coil, the magnetic field environment of the superconducting coil, and the like (i.e., obtaining constraint conditions).
Those skilled in the art will appreciate that the specific means for waveform simplification described above are well known in the art and will not be described in detail herein.
In the embodiment of the invention, as an important point, how to obtain the magnetic field amplitude and the function of the magnetic field change rate and time required by the low-temperature superconducting coil alternating current loss calculation is to reasonably use the H equation (the purpose is not to obtain parameters such as electric field intensity and the like the high-temperature superconducting magnet alternating current loss calculation) to obtain the magnetic field of the low-temperature superconducting coil model and the function of the magnetic field change rate and time, and in addition, a specific magnetic field amplitude is determined as a parameter for calculation based on the obtained parameters. By applying the configuration mode, not only the key parameter required for calculating the alternating current loss of the low-temperature superconducting coil is obtained, but also the accuracy of the key parameter is ensured.
By applying the method for calculating the alternating current loss of the low-temperature superconducting magnet provided by the embodiment of the invention, the model of the low-temperature superconducting coil to be measured is established, and the constraint condition of the model is provided, and on the basis, the method has the key points that: reasonably extracting and calculating used intermediate parameters through an H equation, namely obtaining the magnetic field amplitude of each turn of folded wire of the low-temperature superconducting coil and a function of the magnetic field change rate and time, and selecting a specific magnetic field amplitude in a period to calculate the critical current and the critical current density; and finally accurately acquiring the alternating current loss based on the alternating current loss. The method provided by the embodiment of the invention can accurately calculate the coupling loss and the hysteresis loss of the low-temperature superconducting magnet under any current and any magnetic field respectively and calculate the total alternating current loss of the low-temperature superconducting magnet only by a designed finite element simulation method, does not need any other external resource, has low cost, is simple and easy to implement, can simultaneously divide the alternating current loss of the low-temperature superconducting magnet under different input conditions by parallel calculation, and has engineering application prospect.
As an embodiment of the present invention, in order to obtain any magnetic field amplitude and any magnetic field change rate of the model, the obtaining of the magnetic field amplitude and the magnetic field change rate of the model at any time based on the H equation includes: acquiring the magnetic field intensity of the model at any moment based on the H equation; obtaining the magnetic field amplitude and the magnetic field change rate at any moment based on the magnetic field intensity at any moment; wherein the H equation is shown as follows:
wherein HxAnd HyThe components in the x direction and the y direction of the magnetic field intensity of the superconducting coil model are respectively, and E and J are respectively the electric field distribution and the current density distribution of the superconducting coil model; mu.s0And murVacuum magnetic conductivity and relative magnetic conductivity are respectively adopted; n is the value of n of the superconducting strand and is a constant.
Through whichThe configuration mode is that only partial differential equation module in COMSO L software is needed to be used for solving the magnetic field of the low-temperature superconducting coil model through the formulas (1) to (3), wherein the formula (1) and the formula (2) are Maxwell equation sets, the formula (3) is constitutive equation of the superconducting material, and H isxAnd HyThe distributions of the magnetic field amplitude and the change rate of the superconducting coil at any one time can be obtained from the equations (1) to (3).
As an embodiment of the present invention, the maximum value of the magnetic field amplitude of each turn of the superconducting strand of the model in one period and the function of the magnetic field change rate and time are obtained based on the magnetic field amplitude and the magnetic field change rate, specifically:
1) determining an extraction point of any superconducting strand, wherein the extraction point is the closest point on the superconducting strand to the symmetry axis of the model;
2) and obtaining the maximum value of the magnetic field amplitude at the corresponding extraction point of each turn of the superconducting strand in one period and the function of the magnetic field change rate and time.
In the embodiment of the present invention, specifically, the model established in step 1 is a COMSO L two-dimensional axisymmetric model, and the model is simulated to obtain the distribution of the magnetic field amplitude and the change rate of the superconducting coil at any time, that is, to obtain a magnetic field cloud diagram of the superconducting coil, because the number of turns of the strand of the superconducting coil is large, and the subsequent calculation difficulty is very large after separately calculating the magnetic field distribution at any position of each turn of the strand, the embodiment of the present invention selects an approximate manner to obtain the extraction point of any strand on the basis of the magnetic field cloud diagram, as shown in fig. 2, 1 to N in the diagram represent N turns of the strand, as being equivalent to a two-dimensional model, as an embodiment, each strand may be regarded as a rectangle, for example, for the strand 1, the closest point to the symmetry axis may be taken as an extraction point (for example, the lower left corner point), and after the position is determined, only the maximum magnetic field amplitude and the function of the magnetic field amplitude and the magnetic field change rate in the period of the corresponding strand at the position may be obtained (the function of the magnetic field amplitude and the magnetic field change rate as the function of the corresponding time of.
Through the configuration mode, the key point is how to obtain the maximum value of the magnetic field amplitude of each turn of superconducting strand of the model in each period and the function of the magnetic field change rate and time.
For example: the maximum value of the magnetic field amplitude of each turn of the strand can be expressed as: b ismax1，Bmax2，Bmax3…BmaxNAnd N is the number of turns of the superconducting coil. The magnetic field change rate of each turn of the strand in a period is related to time by N curves, each curve represents the change of the magnetic field change rate of a specific turn of the strand in a period, and therefore the magnetic field change rate of each turn of the strand in a period is related to time by: dB1/dt，dB2/dt，dB3/dt… dBN/dt。
As an embodiment of the present invention, in order to obtain the hysteresis loss, it is further required to obtain parameters including critical current and critical current density of the superconducting strand, and the critical current and critical current density of each turn of the superconducting strand are obtained according to the maximum value of the magnetic field of each turn of the superconducting strand of the model in one period, which is specifically:
for NbTi superconducting strands, determining the relationship between the critical current density of each turn of superconducting strand and the magnetic field and the temperature; or, for Nb3The Sn superconducting strand determines the relation between the critical current density of each turn of superconducting strand and the magnetic field, the temperature and the stress;
and calculating the critical current density and the critical current of each turn of the superconducting strand based on the determined relation.
In the embodiment of the invention, the critical current density and the critical current of the strand can be accurately obtained by determining the relationship between the type of the strand and corresponding factors according to different strand types and resolving the critical current density and the critical current of the strand by researching and analyzing the maximum magnetic field of each turn of superconducting strand based on a model in one period (the magnitude of the magnetic field is the key for calculating the critical current density), and the engineering coverage is ensured.
As an embodiment of the present invention, in order to further ensure the accuracy of the critical current density and critical current, the superconducting strand is an NbTi superconducting strand, and the relationship between the critical current density and the magnetic field and the temperature of each turn of the superconducting strand is as follows:
wherein, C0，BC20，TC0α, gamma are the first, second, third, fourth, fifth and sixth intrinsic parameters respectively associated with the NbTi superconducting strand structure, T0The temperature at which the low temperature superconducting coil is located; b is the maximum value of the magnetic field of each turn of the superconducting strand of the model in one period; j. the design is a squarec(B,T0) Is the critical current density, Jc(B,T0) The ratio of the critical current to the cross-sectional area S of the superconducting strand is defined as the critical current.
In the embodiment of the present invention, when the superconducting strand is an NbTi superconducting strand, the critical current density J of the superconducting strand as shown above is providedc(B,T0) The accuracy of the critical current density can be further ensured by a fitting relation between the critical current density and the magnetic field and the temperature, wherein for the NbTi superconducting strands, the relation between the critical current density of each turn of the superconducting strand and the magnetic field and the temperature is determined, the magnetic field refers to B, and the temperature refers to T0Usually, 4.2K may be taken.
As an embodiment of the present invention, in order to obtain the total coupling loss, obtaining the total coupling loss according to a function of a magnetic field change rate of each turn of the superconducting strand of the one-period model and time and a coupling time constant of the corresponding superconducting strand, includes:
obtaining the coupling time constant of each turn of the superconducting strand as shown in the following formula:
obtaining the coupling loss of any superconducting strand in a period based on the coupling time constant of each turn of superconducting strand and the function of the magnetic field change rate and the time of each turn of superconducting strand of the model in the period, as shown in the following formula:
obtaining the total coupling loss of the superconducting coil based on the coupling loss of any superconducting strand as follows:
Qct＝∑Qci×SLi(10)
wherein θ is a coupling time constant; dBiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; mu.s0Is the vacuum magnetic permeability and is a constant; p is the torque length of the superconducting strand; rhotIs the effective lateral resistivity of the copper matrix; qciCoupling loss of the ith turn of superconducting strand; qctTotal coupling loss of superconducting coil, cross-sectional area of superconducting strand, LiIs the length of the ith turn of superconducting strand; t is the cycle duration.
In the embodiment of the invention, when the coupling time constant of each turn of the superconducting strand is obtained, the internal parameters of the superconducting strand also need to be measured. Specifically, because the internal structure of the superconducting strand is complex, the internal structure of the superconducting strand needs to be analyzed before calculating the coupling time constant and the coupling loss, and the measurement of the internal parameters of the superconducting strand includes: residual resistivity RRR, torque length of the superconducting strand, copper-to-super ratio, effective transverse resistivity of the copper matrix, diameter and number of the superconducting filaments, and the like.
By applying the configuration mode, on the basis of the coupling time constant of each turn of superconducting strand and the magnetic field change rate and time function (key parameters) of each turn of superconducting strand in one period, the coupling loss of each turn of the strand in one period can be obtained through a formula (9), and on the basis, the total coupling loss of the low-temperature superconducting coil can be obtained through a formula (10).
As an embodiment of the present invention, in order to obtain the total hysteresis loss, obtaining the total hysteresis loss according to the critical current, and the critical current density, and the function of the magnetic field change rate of each turn of the superconducting strand of the one-period internal model, includes:
obtaining the hysteresis loss of any superconducting strand in a period, as shown in the following formula;
obtaining the total hysteresis loss of the superconducting coil based on the hysteresis loss of any superconducting strand in one period as follows:
Qht＝∑Qhi×SLi(14)
wherein Q ishiHysteresis loss, dB, of the ith turn of superconducting strandiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; i isciAnd JciCritical current and critical current density of the ith turn of superconducting strand under corresponding magnetic fields respectively; qhtTotal hysteresis loss of superconducting coil, S cross-sectional area of superconducting strand LiThe length of the ith turn of superconducting strand.
By applying the configuration mode, on the basis of the critical current and the critical current density of each turn of superconducting strand and the magnetic field change rate and time function (key parameters) in one period, the hysteresis loss of each turn of the strand in one period can be obtained through a formula (13), and on the basis, the total hysteresis loss of the low-temperature superconducting coil can be obtained through a formula (14).
In order to further understand the method for calculating the ac loss of the low-temperature superconducting coil of the present invention, as shown in fig. 1, the following embodiments are described:
the first step is as follows: the electrical and structural parameters of the superconducting coils and the strands are determined,
specifying the type of superconducting strand for superconducting coils: NbTi strand or Nb3Sn strands, the self-field critical current and the value of n of the strands, and the structural parameters of the coil: the number of turns of the coil is such that,inter-turn insulation, coil dimensions, and the like;
the second step is that: analyzing the current and magnetic field waveforms of the superconducting coil to obtain constraint conditions;
in the third step, COMSO L is modeled,
modeling in finite element analysis software COMSO L according to the electrical parameters and the structural parameters of the superconducting strands and coils in the first step;
the fourth step: setting the constraint conditions includes: the function of the current and the magnetic field,
according to the current waveform and the magnetic field waveform simplified in the second step, the current waveform and the magnetic field waveform are expressed by functional expressions and are input into COMSO L;
the fifth step: solving the magnetic field and the magnetic field change rate of the superconducting coil,
the distribution of the magnetic field of the superconducting coil along with time and the change rate of the magnetic field along with time can be solved based on an H equation by utilizing a partial differential equation module in COMSO L software, and the formula for solving the magnetic field of the superconducting coil through the H equation is as follows:
wherein HxAnd HyThe components in the x direction and the y direction of the magnetic field intensity of the superconducting coil in the two-dimensional model are respectively, and the two components are independent; e and J are electric field distribution and current density distribution of the superconducting coil respectively; mu.s0And murRespectively vacuum magnetic conductivity and relative magnetic conductivity which are constants; n is the n value of the superconducting strand and is a constant; hxAnd HyCan be derived from equations (1), (2) and (3), and therefore the distribution of the magnetic field amplitude and rate of change of the superconducting coil at any one time can also be derived;
and a sixth step: the magnetic field and the rate of change of the magnetic field,
taking the maximum value of the magnetic field amplitude of each turn of the strand of the superconducting coil in one period and the relationship between the magnetic field change rate of each turn of the strand in one period and time, wherein the maximum value of the magnetic field amplitude of each turn of the strand can be expressed as: b ismax1， Bmax2，Bmax3…BmaxNWherein N is the number of turns of the superconducting coil; the magnetic field change rate of each turn of the strand in a period is related to time by N curves, each curve represents the change of the magnetic field change rate of a specific turn of the strand in a period, and therefore the magnetic field change rate of each turn of the strand in a period is related to time by: dB1/dt，dB2/dt，dB3/dt…dBN/dt；
The seventh step: measuring internal parameters of the superconducting strand;
eighth step: the strand coupling time constant is calculated,
and solving the coupling time constant theta of the superconducting strand on the basis of the internal parameters of the superconducting strand in the seventh step, wherein the calculation formula is as follows:
wherein mu0Is the vacuum magnetic permeability and is a constant; p is the torque length of the superconducting strand; rhotIs the effective lateral resistivity of the copper matrix;
the ninth step: the relation between the critical current density of the superconducting strand and the magnetic field, the temperature and the stress is determined,
critical current density J of NbTi superconducting strandc(B,T0) The fit between the magnetic field and temperature is as follows,
wherein C is0,BC20,TC0α, gamma is in excess of NbTiThe first, second, third, fourth, fifth and sixth intrinsic parameters (fitting parameters) related to the guide strand structure, the values of different types of NbTi superconducting strands are also different, and the NbTi superconducting strands can be obtained by measuring the short sample calibration of the NbTi strands;
the tenth step: calculating the critical current and critical current density of the superconducting strand,
critical current density Jc(B,T0) Can be derived from the equations (5) and (6), parameter C0,BC20,TC0α, γ can be obtained by calibrating the measured NbTi strand short sample, and B in formulas (5) and (6) is B in the sixth stepmax，T04.2K, critical current density J of the superconducting strandc(B,T0) The ratio of the cross-sectional area S of the folded yarn is Ic(B,T0)；
The eleventh step: the calculation of the coupling loss of each turn of the strand,
the calculation formula of the coupling loss of the low-temperature superconducting alternating current loss is as follows:
wherein, PcIs the coupling loss power (W/m) of the superconducting strand in unit volume3) Theta is a coupling time constant given by the eighth step, dB/dt is a magnetic field change rate, and the coupling loss Q of the low-temperature superconducting strand per unit volume in one period isc(J/m3A cycle) is PcIntegration over one period:
therefore, the coupling loss of the ith turn of the low-temperature superconducting strand is as follows:
wherein QciIs the coupling loss of the ith turn of superconducting strand, dBiThe dt is the magnetic field change rate of the ith (i is 1,2,3 … N) turn of the strand in one period, and the i is the second turn of the strandObtaining the final product;
the twelfth step: the calculation of the coupling loss of the superconducting coil,
the eleventh step determines the coupling loss of the low temperature superconducting strands per unit volume, and thus the total coupling loss Q of the superconducting coilctComprises the following steps:
Qct＝∑Qci×SLi[J/cycle](10)
wherein S is the cross-sectional area of the superconducting strand, LiIs the length of the ith turn of the strand;
the thirteenth step: the calculation of the hysteresis loss of each turn of the strand,
the hysteresis loss calculation formula of the low-temperature superconducting alternating current loss is as follows:
wherein P ishIs the hysteresis loss power (W/m) of the superconducting strand in unit volume3)，IcAnd JcCritical current and critical current density for the superconducting strand, given by the ninth step, deffIs the effective diameter of the superconducting filament, constant, ItIs the effective value of the current passed by the superconducting coil in one period, and dB/dt is the magnetic field change rate. Then the hysteresis loss Q of the unit volume of the low temperature superconducting strand in one periodh(J/m3A cycle) is PhIntegration over one period:
therefore, the hysteresis loss of the ith turn of the low-temperature superconducting strand is as follows:
wherein QhiHysteresis loss, dB, of the ith turn of superconducting strandiThe/dt is the magnetic field change rate of the ith (I is 1,2,3 … N) turn of the yarn in one period, and is obtained by a sixth step, IciAnd JciAre respectively the ith turnCritical current and critical current density of the strand under corresponding magnetic fields;
the fourteenth step is that: the calculation of the hysteresis loss of the superconducting coils,
the thirteenth step determines the hysteresis loss of the low temperature superconducting strands per unit volume, and thus the total hysteresis loss Q of the superconducting coilhtComprises the following steps:
Qht＝∑Qhi×SLi[J/cycle](14)
wherein S is the sectional area of the superconducting strand, and L i is the length of the ith turn of the strand;
the fifteenth step: the calculation of the ac losses of the superconducting coils,
because the ac loss of the superconducting coil includes coupling loss and hysteresis loss, the ac loss Qt of the superconducting coil is:
Qt＝Qct+Qht[J/cycle](15)。
based on the embodiment, to further illustrate the feasibility of the calculation method of the embodiment of the present invention, the embodiment of the present invention calculates an example coil coupling loss and hysteresis loss based on the calculation method, and compares the calculation result with the result of calorimetry measurement. The example coil is a poloidal field coil (PF) in EAST tokamak apparatus, which contains 14 small coils, PF1, PF2 … PF14, the electrical and structural parameters of the strands and coils are detailed in EAST apparatus manual. Assuming that the PF coil is not affected by an external magnetic field, only current is passed, and the current waveforms thereof are shown in fig. 3, wherein the current waveforms of PF9 and PF10 are consistent with those of PF7 and PF8, respectively, and thus are not shown in the figure.
Fig. 4 is a comparison between results of calorimetry-based tests and results obtained by the calculation method according to the embodiment of the present invention, where the abscissa is the number of the PF coil, and the ordinate is the total ac loss of the corresponding PF coil under the corresponding current waveform. In the figure, the circular marks and the square marks represent the results of the calorimetry measurements and the calculation methods according to the examples of the present invention, respectively. As can be seen from fig. 4, the total ac losses of the PF coils with different numbers are not consistent because the PF coils with different numbers have different structures and different current waveforms, but the measurement result of the ac loss of the PF coil with the same number is substantially consistent with the amplitude of the calculation result, strictly speaking, the measurement result is slightly larger than the calculation result, because the calorimetry is to calculate the loss of the superconducting coil based on the latent heat of the liquid helium by measuring the volatilization amount of the liquid helium, and the loss is not only due to the electromagnetic loss generated by the flow of the superconducting coil, but also due to the heat leakage of the magnet system itself and the mechanical loss generated by the mechanical vibration after the superconducting coil is excited, and the like, so the measured ac loss is the total calorimetry loss; the calculation result only considers the electromagnetic loss of the superconducting coil, namely the hysteresis loss and the coupling loss, but in the EAST device with better heat insulation performance, the alternating current loss of the superconducting coil accounts for most of the total loss. Therefore, the comparison of the results shows that the ac loss result obtained by the calculation method according to the embodiment of the present invention is substantially consistent with the actual situation, which indicates that the calculation method according to the embodiment of the present invention is feasible.
Compared with the prior art, the embodiment of the invention has at least the following advantages:
1. according to the embodiment of the invention, the coupling loss and the hysteresis loss of the complex low-temperature superconducting magnet are respectively calculated from the theoretical calculation and simulation angles, so that the alternating current loss of the low-temperature superconducting magnet can be obtained only by software simulation without performing complicated tests in a liquid helium environment, the analysis time is short, the cost is low, and detailed data support is provided for the research of the superconducting alternating current loss;
2. the embodiment of the invention analyzes the magnetic field size and the distribution of the magnetic field change rate of the low-temperature superconducting coil based on the H equation, namely the embodiment of the invention reasonably extracts the magnetic field amplitude and the magnetic field change rate of each turn of the folded wire of the low-temperature superconducting coil by a finite element simulation method, the method can cover the actual alternating current loss, and has engineering significance (the quantification of dBN/dt by the H equation is one of the key points of the embodiment of the invention, namely the bridging effect is achieved);
3. according to the embodiment of the invention, a bridge is built among the current or magnetic field waveform of the low-temperature superconducting magnet, the size of the magnetic field inside the superconducting magnet and the distribution of the change rate of the magnetic field, the critical current and the critical current density of the superconducting material, and the coupling loss and the hysteresis loss of the superconducting material, and a theoretical relation is given; determining the numerical relation between the critical current density of the low-temperature superconducting coil and the magnetic field amplitude of the coil through a fitting formula; the required parameters are ensured to be accurately obtained, and the alternating current loss of the low-temperature superconducting magnet is finally accurately obtained.
Spatially relative terms, such as "above … …," "above … …," "above … …," "above," and the like, may be used herein for ease of description to describe one device or feature's spatial relationship to another device or feature as illustrated in the figures. It will be understood that the spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if a device in the figures is turned over, devices described as "above" or "on" other devices or configurations would then be oriented "below" or "under" the other devices or configurations. Thus, the exemplary term "above … …" can include both an orientation of "above … …" and "below … …". The device may be otherwise variously oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly.
It should be noted that the terms "first", "second", and the like are used to define the components, and are only used for convenience of distinguishing the corresponding components, and the terms have no special meanings unless otherwise stated, and therefore, the scope of the present invention should not be construed as being limited.
The above is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and various modifications and changes will occur to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (9)
1. A method for calculating AC loss of a low-temperature superconducting magnet is characterized by comprising the following steps:
establishing a model of a low-temperature superconducting coil to be detected;
setting constraints of the model, wherein the constraints comprise: a current function and a magnetic field function;
solving the model based on the constraint condition to obtain the magnetic field amplitude of the model and the function of the magnetic field change rate and time, comprising the following steps:
(1) acquiring the magnetic field amplitude and the magnetic field change rate of the model at any moment based on an H equation;
(2) acquiring the maximum value of the magnetic field amplitude of each turn of superconducting strand of the model in one period and a function of the magnetic field change rate and time on the basis of the magnetic field amplitude and the magnetic field change rate;
obtaining the critical current and the critical current density of each turn of superconducting strand according to the maximum value of the magnetic field of each turn of superconducting strand of the model in one period;
acquiring total coupling loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period and a coupling time constant of the corresponding superconducting strand; acquiring total hysteresis loss according to a function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period, critical current and critical current density;
and obtaining the model alternating current loss according to the total coupling loss and the total hysteresis loss.
2. The method for calculating the ac loss of the cryogenic superconducting magnet according to claim 1, wherein the obtaining the magnetic field amplitude and the magnetic field change rate of the model at any time based on the H equation comprises: acquiring the magnetic field intensity of the model at any moment based on the H equation; obtaining the magnetic field amplitude and the magnetic field change rate at any moment based on the magnetic field intensity at any moment; wherein the H equation is shown as follows:
wherein HxAnd HyThe components in the x direction and the y direction of the magnetic field intensity of the superconducting coil model are respectively, and E and J are respectively the electric field distribution and the current density distribution of the superconducting coil model; mu.s0And murVacuum magnetic conductivity and relative magnetic conductivity are respectively adopted; n is the value of n for the superconducting strand.
3. The method for calculating the alternating current loss of the low-temperature superconducting magnet according to claims 1-2, wherein the maximum value of the magnetic field amplitude and the function of the magnetic field change rate and time of each turn of the superconducting strand of the model in one period are obtained based on the magnetic field amplitude and the magnetic field change rate, and specifically the method comprises the following steps:
determining an extraction point of any superconducting strand, wherein the extraction point is the closest point on the superconducting strand to the symmetry axis of the model;
and obtaining the maximum value of the magnetic field amplitude at the corresponding extraction point of each turn of the superconducting strand in one period and the function of the magnetic field change rate and time.
4. The method for calculating the ac loss of the cryogenic superconducting magnet according to claim 1, wherein the critical current and the critical current density of each turn of the superconducting strand are obtained according to the maximum value of the magnetic field of each turn of the superconducting strand of the model in one period, specifically:
for NbTi superconducting strands, determining the relationship between the critical current density of each turn of superconducting strand and the magnetic field and the temperature;
or the like, or, alternatively,
for Nb3The Sn superconducting strand determines the relation between the critical current density of each turn of superconducting strand and the magnetic field, the temperature and the stress;
and calculating the critical current density and the critical current of each turn of the superconducting strand based on the determined relation.
5. The method for calculating the AC loss of a cryogenic superconducting magnet according to claim 4, wherein the superconducting strand is NbTi superconducting strand, and the relation between the critical current density and the magnetic field and the temperature of each turn of the superconducting strand is as follows:
wherein, C0，BC20，TC0α, gamma are the first, second, third, fourth, fifth and sixth intrinsic parameters respectively associated with the NbTi superconducting strand structure, T0The temperature at which the low temperature superconducting coil is located; b is the maximum value of the magnetic field of each turn of the superconducting strand of the model in one period; j. the design is a squarec(B,T0) Is the critical current density, Jc(B,T0) The ratio of the critical current to the cross-sectional area S of the superconducting strand is defined as the critical current.
6. The method for calculating the ac loss of the cryogenic superconducting magnet according to claim 1, wherein obtaining the total coupling loss according to the function of the magnetic field change rate of each turn of the superconducting strand of the model within one period and the time and the coupling time constant of the corresponding superconducting strand comprises:
obtaining the coupling time constant of each turn of the superconducting strand as shown in the following formula:
obtaining the coupling loss of any superconducting strand in a period based on the coupling time constant of each turn of superconducting strand and the function of the magnetic field change rate and the time of each turn of superconducting strand of the model in the period, as shown in the following formula:
obtaining the total coupling loss of the superconducting coil based on the coupling loss of any superconducting strand as follows:
Qct＝∑Qci×SLi(10)
wherein θ is a coupling time constant; dBiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; mu.s0Is a vacuum magnetic conductivity; p is the torque length of the superconducting strand; rhotIs the effective lateral resistivity of the copper matrix; qciCoupling loss of the ith turn of superconducting strand; qctTotal coupling loss of superconducting coil, cross-sectional area of superconducting strand, LiIs the length of the ith turn of superconducting strand; t is the cycle duration.
7. The method for calculating the ac loss of the superconducting cryomagnet according to claim 1, wherein obtaining the total hysteresis loss according to the critical current and the critical current density, the function of the magnetic field change rate of each turn of the superconducting strand of the one-period model and the time comprises:
obtaining the hysteresis loss of any superconducting strand in a period, as shown in the following formula;
obtaining the total hysteresis loss of the superconducting coil based on the hysteresis loss of any superconducting strand in one period as follows:
Qht＝∑Qhi×SLi(14)
wherein Q ishiHysteresis loss, dB, of the ith turn of superconducting strandiThe dt is the magnetic field change rate and the time function of the ith turn of the superconducting strand in one period; i isciAnd JciCritical current and critical current density of the ith turn of superconducting strand under corresponding magnetic fields respectively; qhtFor total hysteresis losses of superconducting coilsS is the cross-sectional area of the superconducting strand LiThe length of the ith turn of superconducting strand.
8. The method for calculating the ac loss of the superconducting cryomagnet according to claim 1, wherein the modeling the superconducting cryocoil under test comprises:
acquiring electrical and structural parameters of the low-temperature superconducting coil to be detected, wherein the electrical and structural parameters comprise:
determining the type of the superconducting strand used by the low-temperature superconducting coil to be detected, and the self-field critical current and the n value of the superconducting strand; and the structural parameters of the low-temperature superconducting coil to be detected comprise: the number of turns of the coil, the turn-to-turn insulation and the overall dimensions of the coil;
and establishing a model of the low-temperature superconducting coil to be detected in finite element analysis software according to the electrical and structural parameters of the low-temperature superconducting coil to be detected.
9. The method for calculating the ac loss of the cryogenic superconducting magnet according to claim 1, wherein setting the constraints of the model comprises:
analyzing the waveforms of the current and the magnetic field faced by the low-temperature superconducting coil to be detected;
and expressing the current and magnetic field waveforms or the waveforms after the current and magnetic field waveforms are simplified by a functional expression and inputting the waveforms into finite element analysis software.
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